A367270 → θ+1

OEIS matches ?

• A141222: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0: 1;
  1: 1;
  2: 1,   4;
  3: 1,  12,    9;
  4: 1,  24,   54,    16;
  5: 1,  40,  180,   160,    25;
  6: 1,  60,  450,   800,   375,    36;
  7: 1,  84,  945,  2800,  2625,   756,    49;
  8: 1, 112, 1764,  7840, 12250,  7056,  1372,   64;
  9: 1, 144, 3024, 18816, 44100, 42336, 16464, 2304, 81;

The row polynomials satisfy -(n^2-n-(n^4-2*n^3+n^2)*x)*P(n,x) = -(n^2-n-(n^4-2*n^3-5*n^2+10*n-4)*x-(3*n^4-4*n^3-2*n^2+5*n-2)*x^2)*P(n-1,x) - ((2*n-1)*x-2*n^3*x^2+(2*n^3-2*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n - (n^4 - 2*n^3 - 5*n^2 + 10*n - 4)*x - (3*n^4 - 4*n^3 - 2*n^2 + 5*n - 2)*x^2)*P[n - 1, x] - ((2*n - 1)*x - 2*n^3*x^2 + (2*n^3 - 2*n + 1)*x^3)*D[P[n - 1, x], x])/(-n^2 + n + (n^4 - 2*n^3 + n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - n - (n^4 - 2*n^3 - 5*n^2 + 10*n - 4)*x - (3*n^4 - 4*n^3 - 2*n^2 + 5*n - 2)*x^2)*v[i - 1] - ((2*n - 1)*x - 2*n^3*x^2 + (2*n^3 - 2*n + 1)*x^3)*deriv(v[i - 1], x))/(-n^2 + n + (n^4 - 2*n^3 + n^2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-65f24507891d8ee5
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-65f24507891d8ee5
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-65f24507891d8ee5
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-65f24507891d8ee5

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image euler_polynomial, shift 0 e-65f24507891d8ee5

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-2b43df6c63e34146 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

• narayana → (θ+1)²
• A001263 → (θ+1)²

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023